The interaction between nucleon and charmonia ($J/\psi$ and $\eta_c$) is expected to deepen our understanding of various aspects in nonperturbative QCD ranging from the origin of nucleon mass to $J/\psi$ mass modification in nuclear medium and properties of hidden-charm pentaquark states. Here, we present the low-energy $NJ/\psi$ and $N\eta_c$ interactions based on ($2+1$) flavor lattice QCD simulations with nearly physical pion mass $m_\pi=146$ MeV. The interactions, extracted from the spacetime correlations of the nucleon and charmonium system by using the HAL QCD method, are found to be attractive in all distances and manifest a characteristic long-range tail consistent with the two-pion exchange interaction.
The resulting scattering lengths are around $0.3$ fm, $0.4$ fm and $0.2$ fm for $NJ/\psi$ with spin $3/2$, with spin $1/2$, and $N\eta_c$, respectively.
Our results are orders of magnitude larger than those from the photoproduction experiments assuming the vector meson dominance.